
Kinetic mixing: the kinetic mixing (KM) portal is a natural coupling to be expected when a second \(U(1)\)  symmetry is added, since it is gauge invariant and renormalizable. Originally introduced by [1]}, [2]}, the extra U(1) field - usually dubbed hidden or dark photon - can be a generic feature arising from string compactifications [3]}, [4]}, [5]}. It connects the hidden photon with the hypercharge, but if the hidden field is lightweight, the dominant interaction is with the photon through a kinetic mixing term of the form \(\xi F_{\mu \nu } G^{\mu \nu }\) , with \(F_{\mu \nu }\)  the visible field strength, \(G^{\mu \nu }\)  the hidden sector one and \(\xi \)  a dimensionless parameter, expected to be small. It should be also noted that the supersymmetric extension require both gauge mixing and Higgs portal couplings and hence it is a necessary condition in order to have self-dual equations, see [6]}.

Higgs portal: another renormalizable and gauge invariant interaction was first discussed in [7]} in a proposal of possible extensions of the SM. It connects scalar fields \(\phi \)  (visible sector) and \(\eta \)  (hidden sector) via a term \(\lambda |\phi ^2 \Vert \eta |^2\) , with \(\lambda \)  a dimensionless coupling. Such interaction it is widely known as Higgs portal (HP) because it has been heavily exploited identifying \(\phi \)  as the Higgs field (see for instance [8]} and references therein).

Scalar-gauge effective interactions: corresponds to non-renormalizable terms with mass dimension \(\text{dim}>5\)  mediated by a mass scale \(\Lambda \) . An example of this scalar-gauge interaction (SGI portal) is
\(\mathcal {L}_{\Lambda }=\frac{g^2}{\Lambda ^2}|\phi |^2G_{\mu \nu } G^{\mu \nu },\) 
where \(g\)  is the electric charge in the hidden sector.
A possible way to obtain the SGI term is to include quantum corrections, for instance, originated through one-loop like vector-triangle or heavy charged fermions
[9]}, [10]}, [11]}. In this context, \(\Lambda \)  can be expressed in terms of the charge \(g\)  and HP coupling parameter \(\lambda \) .
On the other hand, a more pragmatic approach is to consider the SGI as an effective portal regardless of its physical origin [12]}. In this article we have considered the latter alternative where \(\Lambda \)  is understood as a free parameter.

